The approach of Park et al. [1]} and [2]} assumes that the robot radius is very small. We also compute the collision probabilities for a robot and an obstacle with radius \(0.05\) m each, where the robot and the obstacle are touching each other (Fig. REF (c)). The obstacle location is also much more certain, with the uncertainty reduced by \(97\%\)  as compared to cases in Fig. REF (a),(b). Actual value obtained using numerical integral is \(14.82\%\) . The probability of collision computed using our approach is \(14.83\%\) , whereas, using the approach in [1]} the computed value is \(0.61\%\)  and the approach in [2]} computes it to be \(0.46\%\) . Thus our approach predicts the configuration to be unsafe. To get a sense of the actual value, we compute the area of the covariance matrix, which is \(6.28 \times 10^{-4}m^2\) . This clearly indicates that \(0.61\%\)  is too small a value and the configuration is not \(0.9-\) safe configuration. Using the approaches in [2]}, [1]} would lead to collision as it predicts the configuration to be safe. Our approach computes the exact probability of collision and outperforms the approaches in [2]}, [1]}.
